Problem: Factor the following expression: $5$ $x^2$ $-11$ $x+$ $6$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(6)} &=& 30 \\ {a} + {b} &=& & & {-11} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $30$ and add them together. The factors that add up to ${-11}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-6}$ and ${b}$ is ${-5}$ $ \begin{eqnarray} {ab} &=& ({-6})({-5}) &=& 30 \\ {a} + {b} &=& {-6} + {-5} &=& -11 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 {-6}x {-5}x +{6} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 {-6}x) + ({-5}x +{6}) $ Factor out the common factors: $ x(5x - 6) - 1(5x - 6) $ Notice how $(5x - 6)$ has become a common factor. Factor this out to find the answer. $(5x - 6)(x - 1)$